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On boundaries of multiply connected wandering domains Markus Baumgartner Christian-Albrechts-Universit at zu Kiel baumgartner@math.uni-kiel.de London, 12 March 2015 M. Baumgartner (University of Kiel) Boundaries of wandering domains

  1. Introduction Motivation Theorem (Baker and Dominguez 2000) Let f be an entire function. If J ( f ) is not connected, then it is not locally connected at any point of J ( f ) . This implies that J ( f ) can not be locally connected at any point if f has a multiply connected wandering domain. Question Are at least the different components of J ( f ) locally connected? Theorem (Bishop 2011) There exists an entire function f with dim H J ( f ) = 1 . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 6 / 22

  2. Introduction Motivation Theorem (Baker and Dominguez 2000) Let f be an entire function. If J ( f ) is not connected, then it is not locally connected at any point of J ( f ) . This implies that J ( f ) can not be locally connected at any point if f has a multiply connected wandering domain. Question Are at least the different components of J ( f ) locally connected? Theorem (Bishop 2011) There exists an entire function f with dim H J ( f ) = 1 . Bishop showed that F ( f ) consists of multiply connected wandering domains which are bounded by recitifiable Jordan curves. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 6 / 22

  3. Introduction Motivation Theorem (Baker and Dominguez 2000) Let f be an entire function. If J ( f ) is not connected, then it is not locally connected at any point of J ( f ) . This implies that J ( f ) can not be locally connected at any point if f has a multiply connected wandering domain. Question Are at least the different components of J ( f ) locally connected? Theorem (Bishop 2011) There exists an entire function f with dim H J ( f ) = 1 . Bishop showed that F ( f ) consists of multiply connected wandering domains which are bounded by recitifiable Jordan curves. We want to show that under suitable conditions every boundary component of a multiply connected wandering domain is a curve or even a Jordan curve and therefore locally connected. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 6 / 22

  4. Results Preparations for the results Results Definition (Inner and outer boundary) Let U ⊂ C be a domain and let a ∈ C \ U . We denote by C ( a , U ) the component of C \ U that contains a . U a 0 M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 7 / 22

  5. Results Preparations for the results Results Definition (Inner and outer boundary) Let U ⊂ C be a domain and let a ∈ C \ U . We denote by C ( a , U ) the component of C \ U that contains a . U a 0 C ( a , U ) M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 7 / 22

  6. Results Preparations for the results Results Definition (Inner and outer boundary) Let U ⊂ C be a domain and let a ∈ C \ U . We denote by C ( a , U ) the component of C \ U that contains a . We call ∂ ∞ U = ∂ C ( ∞ , U ) the outer boundary component of U and for 0 / ∈ U we call ∂ 0 U = ∂ C (0 , U ) the inner boundary component of U . U a 0 C ( a , U ) M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 7 / 22

  7. Results Preparations for the results Results Definition (Inner and outer boundary) Let U ⊂ C be a domain and let a ∈ C \ U . We denote by C ( a , U ) the component of C \ U that contains a . We call ∂ ∞ U = ∂ C ( ∞ , U ) the outer boundary component of U and for 0 / ∈ U we call ∂ 0 U = ∂ C (0 , U ) the inner boundary component of U . U a 0 C ( a , U ) ∂ ∞ U M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 7 / 22

  8. Results Preparations for the results Results Definition (Inner and outer boundary) Let U ⊂ C be a domain and let a ∈ C \ U . We denote by C ( a , U ) the component of C \ U that contains a . We call ∂ ∞ U = ∂ C ( ∞ , U ) the outer boundary component of U and for 0 / ∈ U we call ∂ 0 U = ∂ C (0 , U ) the inner boundary component of U . U a ∂ 0 U 0 C ( a , U ) ∂ ∞ U M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 7 / 22

  9. Results Preparations for the results Results Definition (Inner and outer boundary) Let U ⊂ C be a domain and let a ∈ C \ U . We denote by C ( a , U ) the component of C \ U that contains a . We call ∂ ∞ U = ∂ C ( ∞ , U ) the outer boundary component of U and for 0 / ∈ U we call ∂ 0 U = ∂ C (0 , U ) the inner boundary component of U . We call ∂ 0 U and ∂ ∞ U big boundary components . U a ∂ 0 U 0 C ( a , U ) ∂ ∞ U M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 7 / 22

  10. Results Preparations for the results Definition (Connectivity) Let U ⊂ C be a domain. By c ( U ) we denote the connectivity of U , that is the number of connected components of C \ U . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 8 / 22

  11. Results Preparations for the results Definition (Connectivity) Let U ⊂ C be a domain. By c ( U ) we denote the connectivity of U , that is the number of connected components of C \ U . U M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 8 / 22

  12. Results Preparations for the results Definition (Connectivity) Let U ⊂ C be a domain. By c ( U ) we denote the connectivity of U , that is the number of connected components of C \ U . U c ( U ) = 6 M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 8 / 22

  13. Results Preparations for the results Definition (Connectivity) Let U ⊂ C be a domain. By c ( U ) we denote the connectivity of U , that is the number of connected components of C \ U . For a sequence of domains U n we call c the eventual connectivity of U n if c ( U n ) = c for all large n . U c ( U ) = 6 M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 8 / 22

  14. Results Preparations for the results Definition (Connectivity) Let U ⊂ C be a domain. By c ( U ) we denote the connectivity of U , that is the number of connected components of C \ U . For a sequence of domains U n we call c the eventual connectivity of U n if c ( U n ) = c for all large n . U c ( U ) = 6 Kisaka and Shishikura showed that the eventual connectivity of a multiply connected wandering domain is either 2 or ∞ . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 8 / 22

  15. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  16. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  17. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  18. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  19. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n f m ( K ) K B n + m B n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  20. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  21. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  22. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  23. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . We define the eventual inner and outer connectivity respectively. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  24. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . We define the eventual inner and outer connectivity respectively. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  25. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . We define the eventual inner and outer connectivity respectively. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  26. Results Preparations for the results Theorem (Bergweiler, Rippon, Stallard 2013) Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ) . Then every U n contains an annulus B n such that every compact subset K ⊂ U n is mapped inside B n + m under f m for all large m ∈ N . U n + m U n B n + m B n Definition (Inner and outer connectivity) We call c ( U n ∩ C (0 , B n )) the inner connectivity and c ( U n ∩ C ( ∞ , B n )) the outer connectivity of U n . We define the eventual inner and outer connectivity respectively. BRS showed that the eventual inner and outer connectivity is also either 2 or ∞ . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 9 / 22

  27. Results Main result Theorem 1 Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ). M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 10 / 22

  28. Results Main result Theorem 1 Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ). Suppose that there exists a sequence of positive real numbers ( r n ) n ∈ N as well as α, β > 0 such that for a sequence of annuli C n := { z ∈ C : α r n ≤ | z | ≤ β r n } the following conditions hold: M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 10 / 22

  29. Results Main result Theorem 1 Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ). Suppose that there exists a sequence of positive real numbers ( r n ) n ∈ N as well as α, β > 0 such that for a sequence of annuli C n := { z ∈ C : α r n ≤ | z | ≤ β r n } the following conditions hold: ∂ 0 C n ⊂ U n − 1 , ∂ ∞ C n ⊂ U n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 10 / 22

  30. Results Main result Theorem 1 Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ). Suppose that there exists a sequence of positive real numbers ( r n ) n ∈ N as well as α, β > 0 such that for a sequence of annuli C n := { z ∈ C : α r n ≤ | z | ≤ β r n } the following conditions hold: ∂ 0 C n ⊂ U n − 1 , ∂ ∞ C n ⊂ U n C n +1 ⊂ f ( C n ) M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 10 / 22

  31. Results Main result Theorem 1 Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ). Suppose that there exists a sequence of positive real numbers ( r n ) n ∈ N as well as α, β > 0 such that for a sequence of annuli C n := { z ∈ C : α r n ≤ | z | ≤ β r n } the following conditions hold: ∂ 0 C n ⊂ U n − 1 , ∂ ∞ C n ⊂ U n C n +1 ⊂ f ( C n ) There exists m > β α such that for all z ∈ f − 1 ( C n +1 ) ∩ C n � � � � z · f ′ ( z ) �� z · f ′ ( z ) � < π � � � � � ≥ m and � arg 2 . � � � � f ( z ) f ( z ) � M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 10 / 22

  32. Results Main result Theorem 1 Let f be an entire function with a multiply connected wandering domain U = U 0 . Denote U n = f n ( U ). Suppose that there exists a sequence of positive real numbers ( r n ) n ∈ N as well as α, β > 0 such that for a sequence of annuli C n := { z ∈ C : α r n ≤ | z | ≤ β r n } the following conditions hold: ∂ 0 C n ⊂ U n − 1 , ∂ ∞ C n ⊂ U n C n +1 ⊂ f ( C n ) There exists m > β α such that for all z ∈ f − 1 ( C n +1 ) ∩ C n � � � � z · f ′ ( z ) �� z · f ′ ( z ) � < π � � � � � ≥ m and � arg 2 . � � � � f ( z ) f ( z ) � Then all big boundary components are Jordan curves and ∂ ∞ U n − 1 = ∂ 0 U n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 10 / 22

  33. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  34. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have � � z · f ′ ( z ) �� � � � arg � < ε n � � f ( z ) for all z ∈ f − 1 ( C n +1 ) ∩ C n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  35. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have � � z · f ′ ( z ) �� � � � arg � < ε n � � f ( z ) for all z ∈ f − 1 ( C n +1 ) ∩ C n . Then all big boundary components are rectifiable Jordan curves. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  36. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have � � z · f ′ ( z ) �� � � � arg � < ε n � � f ( z ) for all z ∈ f − 1 ( C n +1 ) ∩ C n . Then all big boundary components are rectifiable Jordan curves. Definition (Eventually big boundary components) Let f be an entire function with a multiply connected wandering domain U . Let Z be a boundary component of U . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  37. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have � � z · f ′ ( z ) �� � � � arg � < ε n � � f ( z ) for all z ∈ f − 1 ( C n +1 ) ∩ C n . Then all big boundary components are rectifiable Jordan curves. Definition (Eventually big boundary components) Let f be an entire function with a multiply connected wandering domain U . Let Z be a boundary component of U . We call Z eventually big if f n ( Z ) is a big boundary component of U n for some n ∈ N . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  38. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have � � z · f ′ ( z ) �� � � � arg � < ε n � � f ( z ) for all z ∈ f − 1 ( C n +1 ) ∩ C n . Then all big boundary components are rectifiable Jordan curves. Definition (Eventually big boundary components) Let f be an entire function with a multiply connected wandering domain U . Let Z be a boundary component of U . We call Z eventually big if f n ( Z ) is a big boundary component of U n for some n ∈ N . Corollary 1 Let Z be an eventually big boundary component of U . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  39. Results Further results Theorem 2 Let ( ε n ) n ∈ N be a summable sequence of positive real numbers. Suppose that in addition to the conditions of theorem 1 we have � � z · f ′ ( z ) �� � � � arg � < ε n � � f ( z ) for all z ∈ f − 1 ( C n +1 ) ∩ C n . Then all big boundary components are rectifiable Jordan curves. Definition (Eventually big boundary components) Let f be an entire function with a multiply connected wandering domain U . Let Z be a boundary component of U . We call Z eventually big if f n ( Z ) is a big boundary component of U n for some n ∈ N . Corollary 1 Let Z be an eventually big boundary component of U . Then Z is a closed (rectifiable) curve. Moreover Z is a (rectifiable) Jordan curve if f j ( Z ) does not contain any critical points for all j ∈ N 0 . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 11 / 22

  40. Results Further results Lemma (Joint work with Rippon and Stallard) Let f be an entire function with a multiply connected wandering domain U . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 12 / 22

  41. Results Further results Lemma (Joint work with Rippon and Stallard) Let f be an entire function with a multiply connected wandering domain U . The eventual inner connectivity of U is 2 if and only if every boundary component of U is eventually big. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 12 / 22

  42. Results Further results Lemma (Joint work with Rippon and Stallard) Let f be an entire function with a multiply connected wandering domain U . The eventual inner connectivity of U is 2 if and only if every boundary component of U is eventually big. One direction of the lemma together with corollary 1 implies the following corollary: M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 12 / 22

  43. Results Further results Lemma (Joint work with Rippon and Stallard) Let f be an entire function with a multiply connected wandering domain U . The eventual inner connectivity of U is 2 if and only if every boundary component of U is eventually big. One direction of the lemma together with corollary 1 implies the following corollary: Corollary 2 Suppose that the eventual inner connectivity of U n is 2. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 12 / 22

  44. Results Further results Lemma (Joint work with Rippon and Stallard) Let f be an entire function with a multiply connected wandering domain U . The eventual inner connectivity of U is 2 if and only if every boundary component of U is eventually big. One direction of the lemma together with corollary 1 implies the following corollary: Corollary 2 Suppose that the eventual inner connectivity of U n is 2. Then all wandering domains, which belong to the orbit of U n , are bounded by a countable number of closed (rectifiable) curves. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 12 / 22

  45. Results Further results Lemma (Joint work with Rippon and Stallard) Let f be an entire function with a multiply connected wandering domain U . The eventual inner connectivity of U is 2 if and only if every boundary component of U is eventually big. One direction of the lemma together with corollary 1 implies the following corollary: Corollary 2 Suppose that the eventual inner connectivity of U n is 2. Then all wandering domains, which belong to the orbit of U n , are bounded by a countable number of closed (rectifiable) curves. We can apply Theorem 1 and both corollaries for Baker’s first example of a wandering domain. This means that every multiply connected wandering domain in Baker’s first example is bounded by a countable number of Jordan curves. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 12 / 22

  46. Proof Idea of the proof of theorem 1 Proof Understanding the setting of theorem 1: M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 13 / 22

  47. Proof Idea of the proof of theorem 1 Proof Understanding the setting of theorem 1: α r n r n β r n C n +1 C n +2 C n C n := { z ∈ C : α r n ≤ | z | ≤ β r n } M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 13 / 22

  48. Proof Idea of the proof of theorem 1 Proof Understanding the setting of theorem 1: α r n r n β r n C n +1 C n +2 C n U n +1 U n − 1 U n C n := { z ∈ C : α r n ≤ | z | ≤ β r n } ∂ 0 C n ⊂ U n − 1 , ∂ ∞ C n ⊂ U n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 13 / 22

  49. Proof Idea of the proof of theorem 1 Proof Understanding the setting of theorem 1: α r n r n β r n C n +1 C n +2 C n f U n +1 U n − 1 U n C n := { z ∈ C : α r n ≤ | z | ≤ β r n } ∂ 0 C n ⊂ U n − 1 , ∂ ∞ C n ⊂ U n C n +1 ⊂ f ( C n ) M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 13 / 22

  50. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  51. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. Define for all k ∈ N Γ k := { z ∈ C n : f j ( z ) ∈ C n + j for all j=1,. . . ,k } . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  52. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. Define for all k ∈ N Γ k := { z ∈ C n : f j ( z ) ∈ C n + j for all j=1,. . . ,k } . � � � z · f ′ ( z ) � ≥ m implies that there are no critical points inside the Γ k . The inequality � � f ( z ) So by the Riemann-Hurwitz-formula all Γ k are topological annuli that are bounded by Jordan curves. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  53. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. Define for all k ∈ N Γ k := { z ∈ C n : f j ( z ) ∈ C n + j for all j=1,. . . ,k } . � � � z · f ′ ( z ) � ≥ m implies that there are no critical points inside the Γ k . The inequality � � f ( z ) So by the Riemann-Hurwitz-formula all Γ k are topological annuli that are bounded by Jordan curves. C n C n +1 C n +2 C n +3 M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  54. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. Define for all k ∈ N Γ k := { z ∈ C n : f j ( z ) ∈ C n + j for all j=1,. . . ,k } . � � � z · f ′ ( z ) � ≥ m implies that there are no critical points inside the Γ k . The inequality � � f ( z ) So by the Riemann-Hurwitz-formula all Γ k are topological annuli that are bounded by Jordan curves. C n C n +1 C n +2 C n +3 Γ 1 f M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  55. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. Define for all k ∈ N Γ k := { z ∈ C n : f j ( z ) ∈ C n + j for all j=1,. . . ,k } . � � � z · f ′ ( z ) � ≥ m implies that there are no critical points inside the Γ k . The inequality � � f ( z ) So by the Riemann-Hurwitz-formula all Γ k are topological annuli that are bounded by Jordan curves. C n C n +1 C n +2 C n +3 Γ 1 Γ 2 f f 2 M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  56. Proof Idea of the proof of theorem 1 We want to show that ∂ ∞ U n − 1 and ∂ 0 U n are both curves that coincide. Define for all k ∈ N Γ k := { z ∈ C n : f j ( z ) ∈ C n + j for all j=1,. . . ,k } . � � � z · f ′ ( z ) � ≥ m implies that there are no critical points inside the Γ k . The inequality � � f ( z ) So by the Riemann-Hurwitz-formula all Γ k are topological annuli that are bounded by Jordan curves. C n C n +1 C n +2 C n +3 Γ 1 Γ 2 f Γ 3 f 2 f 3 M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 14 / 22

  57. Proof Idea of the proof of theorem 1 � � � z · f ′ ( z ) The inequality � ≥ m also implies the following lemma: � � f ( z ) M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 15 / 22

  58. Proof Idea of the proof of theorem 1 � � � z · f ′ ( z ) The inequality � ≥ m also implies the following lemma: � � f ( z ) Lemma There exists ̺ > 1 such that we have f k � ′ ( z ) � ≥ ̺ k · r n + k � � � � � � r n for all k ∈ N and z ∈ Γ k . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 15 / 22

  59. Proof Idea of the proof of theorem 1 � � � z · f ′ ( z ) The inequality � ≥ m also implies the following lemma: � � f ( z ) Lemma There exists ̺ > 1 such that we have f k � ′ ( z ) � ≥ ̺ k · r n + k � � � � � � r n for all k ∈ N and z ∈ Γ k . Therefore, f k is expanding inside Γ k and this implies that f − k : C n + k → Γ k is contracting. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 15 / 22

  60. Proof Idea of the proof of theorem 1 � � � z · f ′ ( z ) The inequality � ≥ m also implies the following lemma: � � f ( z ) Lemma There exists ̺ > 1 such that we have f k � ′ ( z ) � ≥ ̺ k · r n + k � � � � � � r n for all k ∈ N and z ∈ Γ k . Therefore, f k is expanding inside Γ k and this implies that f − k : C n + k → Γ k is contracting. We parametrise now ∂ 0 Γ k and ∂ ∞ Γ k as curves by γ 0 k and γ ∞ respectively. k M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 15 / 22

  61. Proof Idea of the proof of theorem 1 � � � z · f ′ ( z ) The inequality � ≥ m also implies the following lemma: � � f ( z ) Lemma There exists ̺ > 1 such that we have f k � ′ ( z ) � ≥ ̺ k · r n + k � � � � � � r n for all k ∈ N and z ∈ Γ k . Therefore, f k is expanding inside Γ k and this implies that f − k : C n + k → Γ k is contracting. We parametrise now ∂ 0 Γ k and ∂ ∞ Γ k as curves by γ 0 k and γ ∞ respectively. k Thereby, one has to check that the parametrisations are compatible with each other. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 15 / 22

  62. Proof Idea of the proof of theorem 1 � � � z · f ′ ( z ) The inequality � ≥ m also implies the following lemma: � � f ( z ) Lemma There exists ̺ > 1 such that we have f k � ′ ( z ) � ≥ ̺ k · r n + k � � � � � � r n for all k ∈ N and z ∈ Γ k . Therefore, f k is expanding inside Γ k and this implies that f − k : C n + k → Γ k is contracting. We parametrise now ∂ 0 Γ k and ∂ ∞ Γ k as curves by γ 0 k and γ ∞ respectively. k Thereby, one has to check that the parametrisations are compatible with each � �� � z · f ′ ( z ) � < π other. Here � arg 2 ensures that the curves are not distorted too much � � f ( z ) under iteration. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 15 / 22

  63. Proof Idea of the proof of theorem 1 ∂ ∞ U n − 1 ∂ 0 U n ∂ ∞ C n ∂ 0 C n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 16 / 22

  64. Proof Idea of the proof of theorem 1 ∂ ∞ U n − 1 ∂ 0 U n γ 0 γ ∞ 1 1 ∂ ∞ C n ∂ 0 C n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 16 / 22

  65. Proof Idea of the proof of theorem 1 ∂ ∞ U n − 1 ∂ 0 U n γ 0 γ 0 γ ∞ γ ∞ 1 2 2 1 ∂ ∞ C n ∂ 0 C n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 16 / 22

  66. Proof Idea of the proof of theorem 1 ∂ ∞ U n − 1 ∂ 0 U n γ 0 γ 0 γ 0 γ ∞ γ ∞ γ ∞ 1 2 3 3 2 1 ∂ ∞ C n ∂ 0 C n M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 16 / 22

  67. Proof Idea of the proof of theorem 1 ∂ ∞ U n − 1 ∂ 0 U n γ 0 γ 0 γ 0 γ γ ∞ γ ∞ γ ∞ 1 2 3 3 2 1 ∂ ∞ C n ∂ 0 C n Then we use that f − k is contracting to show that the curves γ 0 k and γ ∞ converge k uniformly to the same curve γ with � trace( γ ) = Γ k . k ∈ N M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 16 / 22

  68. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  69. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  70. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  71. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . Thus a theorem of Sch¨ onflies yields that γ is in fact a Jordan curve. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  72. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . Thus a theorem of Sch¨ onflies yields that γ is in fact a Jordan curve. This proves theorem 1. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  73. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . Thus a theorem of Sch¨ onflies yields that γ is in fact a Jordan curve. This proves theorem 1. � � �� z · f ′ ( z ) � < π In the proof of theorem 1 we used that � arg 2 bounds the distortion � � f ( z ) of the curves under iteration. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  74. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . Thus a theorem of Sch¨ onflies yields that γ is in fact a Jordan curve. This proves theorem 1. � � �� z · f ′ ( z ) � < π In the proof of theorem 1 we used that � arg 2 bounds the distortion � � f ( z ) of the curves under iteration. � �� � z · f ′ ( z ) In order to prove theorem 2 we exploit that � arg � < ε n ensures that the � � f ( z ) curves are only distorted by a very small amount under iteration. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  75. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . Thus a theorem of Sch¨ onflies yields that γ is in fact a Jordan curve. This proves theorem 1. � � �� z · f ′ ( z ) � < π In the proof of theorem 1 we used that � arg 2 bounds the distortion � � f ( z ) of the curves under iteration. � �� � z · f ′ ( z ) In order to prove theorem 2 we exploit that � arg � < ε n ensures that the � � f ( z ) curves are only distorted by a very small amount under iteration. Thus we are able to show that the curves are close to circles. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

  76. Proof Idea of the proof of theorem 1 and 2 By positioning of C n to U n − 1 and U n we have ∂ ∞ U n − 1 = trace( γ ) = ∂ 0 U n . Now we have that all big boundary components are curves, so it remains to show that they are Jordan curves. Since ∂ ∞ U n − 1 and ∂ 0 U n are curves and therefore locally connected, every point on trace( γ ) is accessible in U n − 1 and in U n . Thus a theorem of Sch¨ onflies yields that γ is in fact a Jordan curve. This proves theorem 1. � � �� z · f ′ ( z ) � < π In the proof of theorem 1 we used that � arg 2 bounds the distortion � � f ( z ) of the curves under iteration. � �� � z · f ′ ( z ) In order to prove theorem 2 we exploit that � arg � < ε n ensures that the � � f ( z ) curves are only distorted by a very small amount under iteration. Thus we are able to show that the curves are close to circles. Therefore, γ is itself close to a circle and hence rectifiable. M. Baumgartner (University of Kiel) Boundaries of wandering domains London, 12 March 2015 17 / 22

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